| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s + 15-s + 16-s − 17-s + 18-s + 19-s + 20-s − 21-s + 22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 28-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s − 7-s + 8-s + 9-s + 10-s + 11-s + 12-s − 13-s − 14-s + 15-s + 16-s − 17-s + 18-s + 19-s + 20-s − 21-s + 22-s − 23-s + 24-s + 25-s − 26-s + 27-s − 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1489 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1489 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.113777007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.113777007\) |
| \(L(1)\) |
\(\approx\) |
\(2.971215055\) |
| \(L(1)\) |
\(\approx\) |
\(2.971215055\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1489 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.35621332536157127908499830809, −20.25515041340212609362598231175, −19.43694964673007300365888647467, −18.65379147284394631665765962311, −17.51421267814970108665610653496, −16.694906696204681299693854552152, −15.95735048207068725790657958084, −15.10614698092417139820927676940, −14.4876333249964467162237974242, −13.66279975950504021678929359905, −13.40516738902663098399194751179, −12.48440862335827696689928064364, −11.82092325829139787223567658229, −10.55803998078251209306615836928, −9.61835037331963764640095324633, −9.4545134200532925045599528635, −8.14839825510851237008687279891, −7.02956801881159813967355891039, −6.61175570402724910785197918267, −5.70902868790173809656170960965, −4.648984848935811959619792096933, −3.8318517245982906721893657497, −2.94265203187744273553172129335, −2.29675944691034466483765636210, −1.41969577883903537687779541398,
1.41969577883903537687779541398, 2.29675944691034466483765636210, 2.94265203187744273553172129335, 3.8318517245982906721893657497, 4.648984848935811959619792096933, 5.70902868790173809656170960965, 6.61175570402724910785197918267, 7.02956801881159813967355891039, 8.14839825510851237008687279891, 9.4545134200532925045599528635, 9.61835037331963764640095324633, 10.55803998078251209306615836928, 11.82092325829139787223567658229, 12.48440862335827696689928064364, 13.40516738902663098399194751179, 13.66279975950504021678929359905, 14.4876333249964467162237974242, 15.10614698092417139820927676940, 15.95735048207068725790657958084, 16.694906696204681299693854552152, 17.51421267814970108665610653496, 18.65379147284394631665765962311, 19.43694964673007300365888647467, 20.25515041340212609362598231175, 20.35621332536157127908499830809
